From a proof I have reason to believe the following inequality holds
where p signifies summing only over primes. But I cannot prove it.
(The full argument used is: )
If I understand correctly, you are trying to bound the number of distinct prime divisors for a number by ? If this is correct, then you can find a loose bound on the number of distinct prime divisors.
Take such that . Then, can have at most distinct prime divisors. Taking the logarithm gives . So in conclusion,
where is prime. This bound is actually sharp: take .